For the Brownian motion on a spherically symmetric Riemannian manifold,
we know a necessary and sufficient condition for recurrence
in terms of the surface area of a ball
(see, e.g., [1]} or [2]}).
Ichihara [3]} established sharp recurrence and transience criteria
for the symmetric diffusion process on \({\mathbb {R}}^d\) 
generated by a strongly local Dirichlet form with unbounded/degenerate coefficient
(see also [1]} for the exposition on this topic,
and [5]} and [1]} for the Brownian motion on a Riemannian manifold).
